Granted, I'm a musician, so take everything I say with a grain of salt. And I don't know what the author's background is.
But I've noticed that the people who tend to have the biggest problem with music theory are engineers. I think it comes from an assumption that music should be learned like engineering, where you start with fundamentals, and then build upon them. And then they're horrified to discover that even the fundamentals are arbitrary and approximate.
Most musicians don't start out with fundamentals. We started by choosing an instrument and somehow learning to play it. For the most part, musicians don't care about temperaments. Equal temperament only applies to a handful of instruments that can be kept in tune long enough for it to matter, and that can play the same note, the same way, twice. An orchestra, jazz band, or solo fiddler, have no temperament in any meaningful sense.
Yes, musical notation is a cluster. But it serves its purpose well enough to be useful to support a symbiosis between the composer and the performer. If any composer chooses to try a bespoke notation, they'll discover that nobody will ever perform their composition. It's like a programming language that nobody likes, but that is supported by a job market and labor pool.
My advice for any engineer wanting to get into music is, get an instrument and learn to play it. Preferably a non-electronic one, so you won't be tempted to turn music into a numbers game.
I'm a professional engineer and amateur musician. I've always made music a bit of an engineering puzzle and I don't necessarily consider that a problem. The thing about engineering music is that once you've established some soft constraints, there's still a ton of room for experimentation. And when you have a good handle on following the rules, breaking them can lead to interesting places too.
I wrestled with temperaments for a bit before realizing that the "engineering rules" of music (at least at the compositional layer) are not dictated by physics but by culture. Equal temperament is in some sense "arbitrary", but because it's what people are used to hearing and performing music in, you have to just accept it as an axiom and move on (unless you're setting out with the express intent of challenging that norm, in which case more power to you).
Also when you delve into world music (particularly from India or the Middle East) you quickly learn that twelve-tone octaves are also arbitrary and that musicians in those regions regularly perform tones that don't fit in that model and have other common musical idioms that western music theory is poorly equipped for because our common music theory originated from a particular western European style from a particular era.
My hope for the future of music notation is that we recognize this and welcome more forms of music theory/notation for different styles of music just like we recognize different strengths of different programming languages.
> But I've noticed that the people who tend to have the biggest problem with music theory are engineers.
My problem with music theory is that in the presentation there is no clear separation of what can be derived from first principles (so, similar to physics theory), and what is based on observations of what people seem to like (so, similar to literary and film theories).
Granted, both physics theory and literary theory are called theories. But they are very different approaches. And then music theory is a mix of both. Some parts can be explained physics-like from psychoacoustic fundamentals, and other parts are like anthropological observations: This is what people seem to like.
And the way it's presented, you don't know which is which.
Much of psychoacoustics was derived from the observation of human hearing. It's not a matter of theorizing from first principles, because the reality on the ground is far more complex. Which is not to say that useful modeling is not possible, just that the scope of such models is necessarily unclear.
As a programmer studying music, I found the "music theory for nerds" article [1] very helpful to get started, and then kept noticing how many general music learning materials for beginners skip the fundamentals/justification, suggesting to just remember many non-arbitrary things that can be deduced, without explaining why they are that way. As the article itself mentions, the hard part is to figure which parts are arbitrary and which ones are not quite so arbitrary, but pretty sure an engineer can get an easy start by studying those bits first.
Playing an instrument is nice and fun too, of course, but I think it's more helpful for intuition rather than actual understanding -- and those are usually best to combine.
I'm a classically trained violinist, and when learning there was a huge sense of "just follow the rules and the sheet music exactly until you get to composing".
In that sense, I think there's a lot of similarities to a lot of engineering practices. There's a straight line to follow on a lot of tasks, adhere to the rules and leave the magic/cleverness to the system architects.
When I switched to guitar, and subsequently discovered the world of prog/math metal, it felt incredibly freeing. However describing that kind of freedom to my friends who adhered to sheet music exclusively was a struggle. Polyrhythms, improvisation, playing by feel were blasphemous to that world.
In that sense I liken it to ratatouille. You must cook the recipe the master chef created exactly, and it's not our place to iterate and experiment.
As a software engineer, I see a lot of the same mindset often. There's a world of engineers who are rigid about sticking to the tried and tested way of doing things. There's nothing wrong with it, in fact it's very valued.
However you need the counterpoint. You need the innovative composers, you need the recipe creators, you need the system designers/architects. Having only one or the other is an imbalance.
I think that's what a lot of engineers get hung up on. Music is taught as rules.
but those rules quickly fall apart when you start having to account for feel, for emotion, for cultural differences etc... Things that aren't able to be captured in rules cleanly.
As a professional classical musician and engineering lead, I agree with you whole heartedly about the parallels between music and engineering.. and I encourage you to take it a step further.
Music, of any genre, is never about reproducing the notes and dynamics accurately. MIDI does that, and the day we can create historically informed MIDI we will have it perfected.
Rather, music is about the spontaneous communication and individuality that occurs WHILE reproducing those notes and dynamics. Genres differ in which dimensions of freedom the musician can use, but not in that fundamental objective. In classical music it's what makes it worthwhile to hear different interpreters of the same piece. It's why it's worth the 5 year waiting list to see Wagner in Bayreuth, when arguably the greatest rendition of The Ring in 100 years (Solti, Chicago, fight me) is available relatively cheaply everywhere.
If that's the case...
1) Encourage your classical friends to break out of the mindset of "cooking the recipe the master chef made exactly." Audiences complain that classical music is sterile and doesn't speak to them, precisely because this mindset is so common among all but the most elite performers. Also, notice that NO elite performers approach their musicianship that way.
2) Notation and theory must have a slightly different purpose than the one you articulated. I believe they're simply there as tools of communication between musicians, to describe recurring patterns we hear. Otherwise we'd be forever saying things like "Beethoven 3, is the one that starts with that thing where it sounds like it's going to end but really doesn't." Calling it a "deceptive cadence" or even a "surprise VI" is just way easier and lets us communicate (and operate) on a higher level of abstraction. It's the same way "dependency injector class" tells you about a given chunk of code, and lets us reason about its place in the larger structure. A great engineer isn't defined by their ability to reproduce a textbook dependency injection class, but rather by their ability to adapt the concept of dependency injection well, in the right places and times.
Very much agreed. Something I found of great interest, especially given my cultural heritage, are alternate forms of musical theory and notation.
I've seen some rather niche notation types that focus on emotion and relative energy while describing notes instead of the usual Italian volume notations alongside the bar notations.
But more interestingly (to me) are things like Indian classical music that doesn't have explicit notation, and instead has a taught relative theory, or other countries like Indonesia that have non octave based scales who's music can only be expressed in western notation by microtones.
Anyway I very much agree that music is a communication language at its core, with music notation existing to aid the spread/consistency of its playback, but it shouldn't stiffle the intention.
> I think that's what a lot of engineers get hung up on. Music is taught as rules.
That's not what I see (as the author of a cross-platform DAW for the last 22 years).
Engineers get hung up on the fact that despite the reality that almost all western (and, realistically) almost all of the world's musical tradition(s) are in fact rooted in clear rules and numerical relationships, music culture wants to use terminology and practice that pretend that it's not.
The simplest example: the major scale. Like all scales, it's just a pattern of intervals. What intervals? Why is the scale not named after the interval series, which would allow you to see how similar it is to the minor scale, and precisely how it differs (ditto for comparisons between any other scale).
Another example: ordinal naming of scale degrees, "the third" etc. These have no consistent meaning other than when you write out the notes of the scale, you count through them. But the third of a minor scale is a different interval from the root than the third of a major scale. If they were named by interval, you'd have consistent naming no matter what the scale is.
There are dozens more examples like this, where consistency in naming would ease many a musician-engineer's mind.
I think your argument is conflating music theory rules with consistency and accessibility.
I agree that the naming and accessibility could be significantly easier. That's just the nature of language though, and music theory is a language. One that's a cumulation of many cultures.
However the rules (in western theory) are very consistent and clear if you've learned your foundations. This is no different than programming too.
If you know the circle of fifths, and the WWHWWWH series of modes, there's a very strong consistency.
Now if we're arguing about why fifths, and why we name our modes after Greek groups rather than something numerical, that's valid. But it's just like the names of the days of the week. Once you know them, it's just an arbitrary term that you internalize.
The rules I'm bemoaning in my post are things like sticking to a fixed time signature or scale, playing perfectly aligned notes. Versus things like polyrhythms and drunk drumming, exploring disharmony or using alternates to octaves.
yes, I understood what you meant by rules. I just don't come across many people complaining about those "rules of practice". If you've listened to reasonable amount of music, you already know that those rules are made to be broken (once learned, maybe).
The problem with the day of the week comparison is that, yes, they are arbitrary and you just learn them and be done with it, but with music the names are just the building blocks for a vocabulary/dialog about a creative process, and so you just don't throw down "play the 5th" the way you might say "it's Wednesday". Instead, that language is interacting with a conceptual process which the language doesn't mirror. If the four components of music are rhythm, timbre, melody and harmony, the last two are all about intervals, and almost none of the music terminology in the west reflects that.
THANK YOU for the difficult work you do working on a DAW. I have enormous respect for your domain, you are "doing god's work" (or noodles' work, or whatever floats your boat) for sure!
I think it's a common mistake, especially in classical music, to think about musical traditions as "rooted in clear rules and numerical relationships." It's analogous to thinking that language is rooted in clear rules and structures.
The truth is just the reverse, the rules, numbers, and structures are rooted in music. They exist to describe an organic, emergent cultural mechanism that is continuously changing. Ask yourself: which came first, music or theory? Language or grammar? There are plenty of musical styles with no formalized structure or numeric relationships, just as there are plenty of languages with no formalized structure or even spelling. Grammar and music theory/notation are fingers pointing at the moon, and we are looking at the finger.
And when we try to engineer systems to help people point at the moon, we should be focused on the finger. Musical structure and numeric relationships are the way people communicate about music, and the tools should speak their language. Unfortunately they have to grapple with the painful inconsistencies in that language and those structures. Computer music comprehension has problems analogous to computer language comprehension - emergent complexity so high it took neural nets to finally achieve real utility.
PS - it's true that Western music has, near the bottom, some relationships that were mathematically derived by none other than Pythagoras, among others. Relationships of fourths, fifths, octaves, and equal temperment all had numeric justification... But that justification was still only there to describe the practice which had already become common, an explanation of "why this sounds harmonious" as well as a proscription for "how to sound harmonious". That some rules are internalized by some generations and broken by others illustrates exactly the problem with the system.
However, I disagree quite a lot with what you've written above. Not really as a matter of historiography - yes, of course, I agree that the history of musical cultures is not the same as the history of mathematics. I agree that musical practice is not rooted in an understanding of numerical relationships. I also agree that the higher level elements of musical composition/improvisation/performance are not sensibly described with math.
However, one half of what defines music as a sensory experience (melody and harmony, the other half being timbre and rhythm/time) is totally, fundamentally, absolutely embedded in the nature of the intervals between pitches. The intervals we use (regardless of any actual absolute pitch selection) and how we use them seem to me to be almost indisputably rooted in the physics/mathematics of the harmonic series even if the development of the musical practice surrounding intervals had been done in complete ignorance of this.
Western musical culture uses a set of terminology and even concepts that obscure the fundamental importance of intervals. The use of scale degrees rather than intervalic terminology means that there is a lack of consistency in naming that isn't necessary (it's really a historical accident), but also a lack of focus on the most important concept for understanding the nature of melody and harmony.
I was really impressed to discover last year that Byzantine musical culture doesn't notate pitch, only intervals. Song melodies are remembered and notated as a series of intervals between the pitches, so that you can start from any note and play something that almost every human being alive will recognize as the same melody (they also do not have octave equivalence, but that's a nother story), even though the actual pitches are totally different. Of course, western musicians can do this too, but consider how much more cognitive load there is in understanding the process when you consider a melody to be made up of specific set of pitches, or a specific set of scale degrees, rather than just a set of intervals.
> But that justification was still only there to describe the practice which had already become common, an explanation of "why this sounds harmonious"
Yes, I agree that the historical ordering is as you describe. We do, however, change our terminology and practice in many other fields as we gain new insight or come to understand better ways of approaching things. I would argue that we could learn from both other musical cultures, and from the numerical/physical understanding of intervals to get to much more rational, consistent terminology and practice when it comes to melody and harmony.
>My advice for any engineer wanting to get into music is, get an instrument and learn to play it. Preferably a non-electronic one, so you won't be tempted to turn music into a numbers game.
Not debating your advice but just saying that some brains work differently.
I play both guitar and piano and also know the "math numbers" of music theory and after decades of learning, I've concluded that the mechanical aspect of hands playing an instrument often limited my understanding of music while the numbers aspect always enhanced it.
I was about 11 years old when I first learned about mathematical relationships of note frequencies from an computer programming book. So to make the computer's sound chip play a C Major arpeggio would be coding something like this in the BASIC language:
I've always liked knowing that octaves are 1/2x or 2x multipliers frequency. And the guitar frets spacing is a visual representation of the 2^(1/12) formula. And because a B1 is 61.7Hz, that's why a 60Hz hum of electricity sounds like a slightly flat low B. And math group theory perspective of chord constructions helped me see how Chopin piano voicings transformed in a coherent way.
For people like me, I'd slightly reword your advice as "My advice for any engineer wanting to get into music [enjoyment feedback loop] is, get an instrument and learn to play it."
If you're engineering-minded like me, the "systems" approach of the numbers behind the music explains it in a way that the fingers will not.
This is exactly correct. Especially software developers who are new to music often approach it like learning a programming language, which is not how musicians think and learn music at all.
Equal temperament only applies to a handful of instruments that can be kept in tune long enough for it to matter, and that can play the same note, the same way, twice.
I'm confused. If I'm not mistaken, equal temperament is the norm for all instruments since Bach.
Any keyboard instrument is tuned that way. Or any instrument with frets. Or any instrument that must be played alonside with one of the previous ones.
If you meant something else, could you ellaborate?
Negative. I’m a classically trained guitarist with degrees in music and physics. You are correct about the frets, but not the strings themselves. Sometimes, you really need the two E strings to be perfectly in tune, sometimes, some piece has an octave on two adjacent strings as a drone, and that needs to be aligned, or in the key of D (with a drop D), you’d probably want the two Ds to be in tune. Whatever you choose, something will be out of tune because of physics . Half steps don’t perfectly align with their equivalents - thirds in particular drive me nuts. It’s just a matter of picking what you hate the least. It’s not uncommon to see professional classical guitarists give a string a micro adjustment during a whole note/rest while they are playing, or tweak tuning between each piece.
OK, I see what you mean, thank you very much. That's actually very interesting for me and explains some inconsistencies that I had attributed to too new/too old strings, badly adjusted bridge or whatnot.
Old strings are a different matter. Strings ideally obey Hook's law for small perturbations and undergo simple harmonic oscillation with standing waves (overtones) that have nodes at the neck and bridge (or fret and bridge). With old strings in particular, the fundamental of the string changes over time (as the string sounds). Audibly so! My mental model is that the "length" of the string is a really fuzzy concept for [worn] strings in motion. Plucking the string deforms it, and the string slowly retracts back to the stable length (at rest) as it dissipates energy. I haven't done any math or seen any, but I don't think its linear, either. You can easily verify this using an electric guitar if you plug it into a tuner thing with a visible meter - the arm will swing around. Also, I haven't done any physics in over a decade because I write software now...
I think that mathematically equal temperament didn't come into use until the modern piano with an iron frame, though I don't know how pipe organs were tuned. At the time of Bach, a keyboard instrument would go out of tune quickly, so a musician had to be able to tune their own instrument quickly, before every performance. Modern piano tuning is a specialized skill.
Temperament methods evolved towards ever-better approximations of equal temperament. They all had names. Bach's "Well Tempered Klavier" was written for an instrument tuned in the Well temperament, and explored the unique tone quality of each key.
String and wind instruments can't be played in tune consistently enough to worry about temperament. And musicians will push notes slightly sharp or flat for emphasis. On the cello, I was taught to slightly stretch whole-step intervals, and squeeze half-step intervals together when playing scales. It made a major scale sound "more major" for instance. Every wind instrument has one or two notes that are known to be bad and must be corrected on the fly, though it's less noticeable on faster passages. The musicians play in tune by listening to one another.
Fretted instruments in Bach's time were an odd situation. Gut strings vary in density along their length, and go out of tune as they age. The viola da gamba had frets, but they were movable and required periodic adjustment.
Even the temperament on a modern steel-stringed guitar is approximate. The compensated bridge is an attempt to correct the scale, but it's not perfect. And it's not systematically tempered, as in, adjacent octaves are not in tune with one another.
I see you're talking about orchestal music that seems to be a world apart from pop music instrumentation, both because there's less use of electronic sound generation and because chords are played yuxtaposing several instruments.
About guitars, I can believe the bridge, or even finger pressure provide adjustement for individual notes. But if you're playing a chord, you just trust the frets, that are uniformly spaced, following the logarithmic scale.
Lots of instruments use "just-in-time intonation". Like the signing voice or playing a violin: the intonation is what you play. The ear prefers each chord to follow just intonation if you really try to be that accurate.
In that sense, your violin's strings might be tuned to match an equal tempered piano, but your own intonation will not (probably) follow any "equal temperament" for the notes you play on the fingerboard, unless you follow an external reference, not your ear.
When you try to play a fifth on a violin, your ear will help you to play it exactly (ratio 1.5), not the equal temperament approximation. Of course, for the fifth the difference is very small, but the same idea applies to the other intervals.
In my understanding and all the more knowldgeable people than me that have taught me music, that's absolutely not correct. Equal temperament is just the water you swim in, for all western music since Bach.
"Natural" tuning by integer fractions would sound very unusual to everybody.
I think you're mixing it up with some other aspect of temperament.
Take the example barbershop's use of justly intonated chords. It doesn't sound unusual, it sounds very clean and pure.
Take the example of Jacob Collier's vocal arrangements, he likes to sing the chords justly intonated too. He has various explanations of this on youtube. Also gifted with an insane ear so that he can clearly sing/explain the difference.
(I don't know much about playing the violin!) Intonation is a living thing - you use your ear, it's a feedback system - listen to your own instrument and those you are playing with.
> Intonation is relative. When playing in an ensemble, care must be taken to blend with the others in the group.
Do you find Barbershop music to sound "very unusual"? The defining feature of the style is just intonation (and microtonal just intonation at that, because a harmonic seventh can't be adequately approximated by 12 equal temperament).
Many issues. The tuning of each instrument is a bag of compromises, especially on instruments where the notes are not entirely independent of one another. Woodwind instruments have fewer keys than notes, so each key has multiple roles. Each octave has its own quirks.
Woodwind instruments have "bad" notes, and I once saw an advertisement for a flute with a "new improved scale." Brass instruments have their main valves, but also little tuning slides with hooks on them so you can work them in and out with your extra fingers while playing. It's fascinating to see someone play tuba, it takes both hands to work the thing properly.
Intonation is a lifelong journey for most musicians. Near the end of his life, they interviewed Ray Brown and asked him if he still needed to practice. He said that he still practiced every day, to improve his intonation. If players sound in tune, it's because of an almost obsessive effort to make that happen.
What I'm saying is you can't begin to talk about temperaments until you can play the same note at the same pitch, in any octave, at least twice. I'm a double bassist, and I'm still working on that, after playing for nearly 40 years.
> My advice for any engineer wanting to get into music is, get an instrument and learn to play it.
The most useful music theory I've ever learned was the stuff that I needed to play jazz guitar: to improvise solos and comp chords. That course of study has continued to pay dividends many years later.
The second most useful was an esoteric but extremely approachable exercise called "arch mapping", where you draw arches on graph paper to represent the form of a piece of music in terms of its subdivisions. For music with a steady tempo and constant time signature, basically anybody can try it: https://williamwieland.midcoip.net/theory/form/archmaps.htm
I also like arch mapping because it explodes the possibilities of what "music theory" is. It can be applied to literally any genre. You don't need to read music. You don't need to know western harmonic theory. You don't even need to know what a major chord is!
As educator for engineers, I would even go so far as to say your approach is good for engineering too.
I had the impression that fundamentals are much easier if you played around with some real world tech.
But to go back to music theory. I think, it helps when you already started a song and you're stuck. If you got a catchy melody, some theory gives you the tools to add fitting harmonies to it, etc.
Funny, I know tons of hobby "musicians" (I use that term loosely here) that are professional engineers. I've always observed that engineers are often more interested in playing music than others.
Perhaps we are observing the same thing still: engineers gravitate towards music but then get stuck because the rules are so arbitrary.
Oh, and speaking of making music a numbers game: check out Tool's Lateralus!
Learning music is more like a language than a discipline in my experience. You can do it the engineers way, but if you follow the way babies learn speech then it becomes a lot more gratifying and things become more intuitive. Victor Wooten had a ted talk on the subject.
Learn by mimicking, then learn structure after you've already had fun just making some noise.
I'm not a musician but have tried many times to learn and I can definitely agree. It has always been frustrating how inadequate explanations coming from musicians are.
This is why I suggest the alternative of "get an instrument and learn to play it," because it bypasses those roadblocks. And if it gets you to the point where it provides you with some enjoyment, so much the better.
Now, you have to choose an instrument. There's a good reason why the guitar remains popular. The learning curve for tone production is extremely gentle, and there's a huge variety of styles that you can enjoy, plus online learning materials. It's portable, affordable, and alcohol powered. ;-) It looks like Fender is moving towards a new business model that revolves around learning to play.
Isn't inadequate here completely subjective though? If you're expecting a set of hard rules, that's what music theory is for. If you're expecting the rest of the picture, that's easy to describe in an emotional sense but hard to form as rules.
Music is a social convention about the same as language. It may be interesting where words or grammar originate, but it’s ultimately not important to communication.
There’s a bit more to this: a “perfect” fifth is not actually two notes differing by a factor of 2^(7/12), but rather by a factor of 1.5. It just so happens that 2^(7/12) is very close to 1.5, which makes it possible to tune a piano so that all fifths and all octaves sound approximately right.
Theoretically, though, there will always be some error: any non-integer power of two is necessarily irrational. Thus, there is no perfect way to divide an octave into equally hz-spaced increments so that you hit the fifth along the way to doing the whole octave.
I believe it's also one of the reasons the G-string is hardest to tune by ear because in equal temperament it's actually the furthest away from the true note.
Yes, between most guitar strings is a perfect 4th interval (where equal temperament ratio [0] and the integer ratio it approximates [1] are quite close to each other) but between the G string and B string is a major 3rd interval (where equal temperament ratio [3] and the integer ratio it approximates [4] are a bit farther from each other).
When using a tuner to tune a guitar, you get the equal temperament normally desired for it's ability to play equally in any key. When tuning by ear, if you are just minimizing the beating you hear (instead of targeting the beating that exists in equal temperament, like good piano technicians are trained to do) you'll end up with integer ratios. Accidentally ending up at the integer ratio of a major third is "farther off" than doing so for a perfect fourth. Typically this manifests as a difficult time tuning the B string though, not the G string, if you go low to high. Solution: tune low to high for the E,A,D,G and tune high to low for the high E,B (doing a perfect octave for the E to E where the equal temperament ratio [4] perfectly matches the integer ratio [5]).
I find it easier not to tune either high to low nor low to high, but use lots of cross-string intervals and a variety of chords. Since my tuning fork is a standard A, I start on the A string, and use that to tune both E strings and the D, then B relative to high E and D relative to A. It’s good to tune G using both octaves from the low E and A string. This way you never tune a string more than 2 steps away from the reference, as opposed to tuning E,A,D,G, which gives you a G string 3 steps away from the reference.
But recently I discovered the most helpful trick, use a fifth from G down to the A string, and also use a fifth from the D to the low E string. Those last two intervals can be the most surprisingly out of tune intervals when using either ear tuning or an electronic tuner.
Does this mean there's actually a scientific reason to tune by ear instead of with an electronic tuner? I feel like I naturally do some of these things, like tuning low to high but then (seemingly arbitrarily) tune the high e against the low E.
Yes, integer ratios sound better than equal temperament ratios. The downside is that you end up with an instrument that has some integer ratios and some disgusting wolf intervals because it's mathematically impossible to have all integer ratios between every possible interval in every possible key you might decide to play in. That problem is what equal temperament solves for, and is a good reason to use an electronic tuner unless you know your songs will sound better without.
Please let me know if I'm understanding. This technique is for tuning open strings and listening, back-and-forth, to the perfect fourth (or major third) interval?
In contrast to matching the same note on the next string?
Yeah, I should have clarified, but at the time of writing, I didn't even realize it (because I'm more of a piano tuner than a guitar tuner so I don't have the luxury of frets that automatically impose equal temperament!) -- yes, this issue occurs when comparing open strings or unfretted harmonics.
That's actually the crux of the issue: if you fret the note to tune the next string, then you achieve equal temperament at the expense of any inaccuracy that comes from all sorts of things like imperfect action height, imperfect bridge tuners, excessive pressure beyond just contacting the fret, sideways motion, etc. which is all avoided by using open strings instead. But then doing open strings by ear doesn't get you equal temperament, it gets you integer ratios instead. Open strings with an electronic tuner gets you the best of both worlds (assuming you want equal temperament).
No, my entire comment is about tuning a typical fretted guitar. Playing a unison as you say, and turning the peg until beating completely disappears, yields integer ratios instead of equal temperament ratios. The result of that can sound nicer than equal temperament for some keys/chords but terrible for other keys/chords.
Ah shoot. I screwed up (only in the comment you're questioning, not my earlier comment per se). Of course the frets are positioned according to equal temperament, so when fretting a note for tuning the next string, indeed you end up at equal temperament when getting the beating down to nothing as you play the unison.
Rather, I'm talking from a point of view of playing two adjacent open strings. Or harmonics thereof (as some people like to tune with 7th fret harmonics and so forth). If you aren't fretting and you minimize beating down to nothing, then you end up with integer ratios.
IIRC harmonics ignore frets. You use the fret to approach the point where it happens, but its exact position is a function of the whole string lenght and it's therefore not equal temperament related.
I think you're right about the "beating"... in general this post has been extremely instructive for me. My teachers were piano players. Either they didn't know better (because they didn't need it) or they didn't bother to explain the fine details to me. Even the books I consulted later don't tell the whole story. That's the magic of the Internet :-)
Correct! Tuning open strings (listening to an interval) and tuning by harmonics (listening to a unison without using frets) both result in a lack of equal temperament if you eliminate the beating entirely. Only good piano tuners are really experienced in landing at the correct beat frequencies without electronic aid.
Whether G is hard or B is hard depends on whether you're going high-to-low or low-to-high, as the issue is the interval between them, so it's just a matter of which one you're using as a known-good reference and which one you're trying to tune. There's an easier solution than taking multiple passes as you play different chords: see my other comment.
Thanks. I understand your approach but what method do you use to tune the high E? I assume you mean that the low E is used to tune the high E but perhaps you pick an E on the G string for this purpose. Is there any "best" E on the low side to use as a reference?
You always need some point of reference to start with, or if you don't have an external reference, you can just go with anything at all. I would typically choose either of the E strings, just to be able to immediately match the other E string to it, and then go inward (up to G and down to B).
I tried it and it works pretty well. One thing that I noticed is I subconsciously have been tuning GBE open because it’s a pain to reach around to the tuning pegs. How does this impact things?
When you play G and B open and tune them to eliminate beating, you're in pretty bad shape because you're making a "just" (integer ratio) major 3rd which is quite far from an equal temperament major 3rd.
When you play B and E open and tune them to eliminate beating, you're in okay shape because you're making a "just" (integer ratio) perfect 4th which is not terribly far from an equal temperament perfect 4th.
See my earlier comment [0] with all the numbers if you want to quantity "pretty bad" vs "okay". In that comment, my initial references to footnotes 3 and 4 should actually be 2 and 3, but it's too late to correct it.
The more interesting question to me, is not what makes a major or minor, that is a simple google search away, there are tons of references about that; the more interesting question in my mind is why do certain harmonic intervals sound more pleasing to humans (perhaps animals as well) than other intervals.
This is almost certainly an oversimplification (e.g. the listener's familiarity with the sounds also matters), but the results match my own perception of consonance and dissonance reasonably accurately. It doesn't address the question of "why", but intuitively it seems that dissonance is associated with sounds that take more mental effort to understand. If the partials line up closely then they are perceived as part of the same sound, and if they are spaced far apart then they are obviously different, but there's an awkward intermediate spacing that's uncertain (the "roughness" on the Plomp and Levelt curve). Dissonant sounds contain a lot of this uncertainty.
If it was as simple as familiarity, then babies would not recognize dissonance
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3735980/
Although I suppose there could be an argument that familiarity began in the womb but that seems like a stretch IMO.
How a chord sounds depends greatly on the timbre of the instrument or instruments. For example, a major or minor triad played on a heavily distorted electric guitar sounds pretty uncomfortable (hence power chords), but played on strings sounds pretty calm.
How a chord sounds also depends greatly on what came before it. For example, in typical common practice period harmony, dominant seventh chords do not sound resolved. On the other hand, many songs in popular music have moments that resolve to a dominant seventh chord.
A distorted guitar distorts the components of the power chord after mixing them. This means that you get intermodulation distortion between different notes. The sum and different tones generated by this distortion form a harmonic series when combined with the original notes.
A power chord is root + fifth, which is a 1:1.5 ratio of frequencies (exactly in just intonation, and very nearly in equal temperament). Intermodulation distortion produces tones of frequency ratio 1.5 + 1 and 1.5 - 1, giving 2 and 0.5. Combine them with the existing tones and you get 0.5:1:1.5:2. This is just the harmonic series 1:2:3:4, but pitched down an octave (all frequencies divided by two). A distorted power chord contains the harmonics of a single note pitched an octave lower, which is why it sounds "heavy".
There are many other harmonics in the notes, so the timbre is much more complicated that that, but the fundamentals are most prominent, so the distorted power chord sounds consonant.
And if you have a guitar with a hexaphonic pickup and distort each string individually you don't get intermodulation distortion between the notes, so you can play much more complicated chords without it sounding dissonant.
...the more interesting question in my mind is why do certain harmonic intervals sound more pleasing to humans...
Unfortunately the first satisfactory answer that I got needed much more knowledge. The foundation is about proximity and dynamics (thinking about scales as a progression, not simultaneous notes).
But it's much better to just accept that major is "happier" than minor when you're starting, and later understand why intuitively.
The classical answer to this question is given in On the Sensation of Tone As a Physiological Basis for the Theory of Music, a long book by Helmholtz (the famous physicist). He attempts to derive all of Western harmonic practice from physiological considerations. It’s probably controversial, but it’s a fascinating book.
I wonder what it is about music that makes people accept these simplifications. I imagine an alternate universe where an article on the optics of color is posted. Someone mentions that all light are frequencies, the color red is such-and-such frequency, blue is whatever Hz, etc. We observe that a color wheel creates some theory of adjacency, compliments of color etc. And some artists use this in their work to enhance the effect. Would an article like that warrant a comment section filled with a bunch of people acting like that gave them sufficient tools to understand the works of Da Vinci, Van Gogh, Picasso?
Pitch, melody and harmony are integral aspects to music but they are by no means sufficient to provide an understanding of music. Just as a purely mathematical description of the optics of color is not sufficient to understand painting. I would argue that the overwhelming majority of music you have enjoyed in your life was written with literally zero regard to these details. In my own opinion, music programs that raise these mathematical details above all others lead to the dullest and least inspired music.
This is an odd post, because the author starts in the right place, harmonics, and then veers off into scale degrees without really answering the question.
The best place to get the answer is Schönberg’s Theory of Harmony, but we can do something quick here.
The main harmonic series of music is what happens when one applies integer multiplication to a given frequency, which we’ll call a root. As the author described, an octave, or the same perceived sound but higher, is a 2x, and a 3x, or 1.5x the previous value, is called a “fifth” due to it being the fifth major scale degree in the twelve tone system described by the author. 4x is 2^2, another octave, and then 5x is the (major) third.
Because all of these integer harmonics are present in the root, though at greatly reduced force, these sounds serve to bolster the natural harmonics of the root.
While it's an interesting derivation, Schoenberg was eager to explain that his book was about teaching practical skills and understanding.
He prefaced his derivation with the statement: "Therefore, whenever I theorize, it is less important whether these theories be right than whether they be useful as comparisons to clarify the object and to give the study perspective."
This is something I find compelling about Schoenberg. He doesn't claim to know more than he does, he's just presenting a useful mental model.
This might just be a word choice thing, I think I took issue with it being labeled the answer -- Schoenberg didn't want it to be considered definitive, but merely useful.
An overarching theme of the book was his philosophy that no eternal aesthetic laws were known to exist, so he was careful not to make definitive claims unless he could really be sure about them.
Chuckle. This is why people can’t stand engineers. (Said with love, but still).
The idea that mhz is even worth mentioning…
Fwiw, huge shout out to https://improviseforreal.com which for me was the happy medium between “major chord is happy” (huh?, whyyy?) and “C0 is 16 mhz”.
Sorry, but I have to say that none of this will make a particle of sense to someone who would be asking such a question.
A rudimentary question is a wonderful an opportunity to distill everything down to a simple understandable concept.
Here's how I would do it:
---
Western music is based on simple frequency ratios. 2:1 is so basic that we call it the same note, but an octave up.
Three pitches with ratios 4:5:6 make a major chord. These are the simplest possible ratios within an octave, so a major chord is the most basic chord, and all other chords are derived from that
>Western music is based on simple frequency ratios.
Western music is based on simple frequency ratios, played by instruments with harmonic timbres (meaning that when you decompose the sound of a single note into sine waves using a Fourier transform, the frequencies of the sine waves are all integer multiples of the lowest frequency sine wave).
This is true for wind and string instruments, but not true with tuned percussion unless you go to special effort to shape the instrument to make it so (as is done by bell makers). Musical styles that use inharmonic timbres, e.g. Indonesian gamelan music, need different tuning to sound best.
And this also happens in Western music. Piano is technically tuned percussion, and the sound of a piano is slightly inharmonic. Pianos are tuned with stretched octave to compensate:
But the harmonic or inharmonic content of the instrument does not address the original question.
And it leads to confusion; putting your first and last sentences together implies that a piano is not a western musical instrument. We know that's not true.
Piano timbre is approximately harmonic. The inharmonicity is subtle enough that you can ignore it and it will still sound okay. Same is true with plucked string instruments. You're still using Western music theory.
Yes, of course. I've tuned pianos, so I'm not disagreeing with that.
I'm just struck by the zenlike simplicity of the original question, "What is a Major Chord?", compared to the random bits of technical jibber-jabber that would confuse the hell out of a poor soul who would dare to ask such a question, both in the linked blog post, and the discussion here.
The motivating question, which I quote at the beginning, was actually: what is a chord? what is a major chord? what is a note? what is a first/fourth/fifth note? is there a 65th note? what is a scale? what is a major scale? what does it mean that a note is of a scale? what does it mean that a chord uses a note? is there a difference between a chord using a note of a scale and not of a scale?
I picked one of them for the title of my post, but I'm trying to answer them all.
I've been a musician since I learned to play the clarinet beginning in 4th grade, almost 57 years ago. During that time, I've learned to play the saxophone, guitar, and the piano. I learned how to read music from the very start and have been totally accustomed to the western notation. Is it the best? I have no idea. It's the only way I know. That said, anybody attempting to design a new, better, notation is facing long odds in it every being widely adopted, even if it can demonstrate some advantages. Trying to persuade the music community would be like trying to convince typists to use the Dvorak keyboard instead of the standard. Yes, some will adopt it and sing its praises but most will not want to relearn how to type.
It's harmony produced the root (any frequency), the major third (5/4 ratio), the 5th (3/2 ratio).
They're approximated in the 12TET scale so that they harmonize across keys and octaves.
I always say I hate musical notation (letters and classical notation) being it so horrifically obscures the mathematical relationships of the 12TET scale. It's literally just approximations of:
Major Scale
Root frequency (C 262hz, whatever)
2: 9/8 root (D)
3: 5/4 root (E)
4: 4/3 root (F)
5: 3/2 root (G)
6: 5/3 root (A)
7: 15/8 root (B)
2x root (C, 524 hz)
2: 9/8 2x root
and so on.
Minor scale is simply the starting the whole thing at the Major 6th instead of at the root.
12TET in comparison to just temperament is slightly harder to explain because of the way it allows harmonies to scale without distorting when changing keys, but there really isn't much more to western music than that.
I'm not much of a musician, but all the explanations about why the major scale is the way is, or octave equivalence, etc. always sound suspiciously similar to the good old "nature abhors a vacuum" [1] to me.
I feel like it has much more to do with convention than music theorists like to confess.
In case some folks have not seen it: along something of the same lines but in very much more depth is Music: A Mathematical Offering by Dave Benson, at https://logosfoundation.org/kursus/music_math.pdf
One thing they do not mention is a stringed instrument creates a standing wave which produces many octaves simultaneously when playing a single note. This sounds considerably different from a pure sine wave and is possible to leverage in a creative way on guitar/guitar like instruments.
Not just many octaves: many harmonics. With an ideal string, these are all multiples of the fundamental frequency, because they each represent a different way of dividing the whole string into equal parts.
As a tentatively more minimalistic explanation, the musical ear perceives specific geometric relationships between frequencies as strong stimuli, and a major chord consists of simultaneously striking frequencies x, x times the cube root of 2, and x times 3/2.
The fact that 7 of the 12 notes in an octave are assigned letters has always bothered me. It's quite arbitrary, we could just as easily have 6 letters + 6 in-between notes, or just 12 discrete letters.
I had similar thoughts and can recommend this video that helped me understand why the sharps/flats exist. It goes through a brief history of western music before harmonization was a thing, and then needing to introduce new notes in order to have perfect 4ths and 5ths.
https://youtu.be/r7aQQQsvxho
It’s not arbitrary. Heptatonic scale is the cornerstone of Western music. Twelve-tone equal temperament is a 18th century invention created to make modulation between heptatonic scales easier.
The seven notes that have their own letters are what’s called diatonal in the Western musical tradition. It turns out that however you divide the octave (12 semitones in the West), you can make a lot of music using only some subset of the pitches (7 in the West) and people gravitate towards those subsets because of their familiarity. The rest, those that in the West are notated with sharps and flats and called chromatic tones – the black keys of a piano – are basically for situations where you need extra expressive power. They literally add "color".
(For simplicity’s sake I assume the Cmaj/Amin scale here. In non-Romance countries we typically teach basics of music using relative solfege aka “movable do”, where notes always have the same names (Do, Re, Mi etc) no matter what the absolute pitch of the tonic is)
The naming makes perfect sense with C Major, or A minor. Because the in-key notes are letters. The weirdness is how we then apply those same names to every other key.
I tend to be very C Major focused when playing piano, and just use the transpose function to change keys. Things are much simpler that way IMO.
Every major scale contains one note with each name (ABCDEFG). For example, Bb major is Bb, C, D, Eb, F, G, A.
If you start with C major (no sharps), and ascend in 5ths (G, D, A, etc.) then you add one sharp to the previous major scale to get the major scale starting on the new note. The added sharp each time is the 7th of the scale. So G major just has F#, D major adds also C#, A major adds also G# etc.
If you start with C major (no flats), and descend in 5ths (equivalently: ascend in 4ths) (F, Bb, Eb, etc.) then you add one flat to the previous major scale to get the major scale starting on the new note. The added flat each time is the 4th of the scale. So F major just has Bb, Bb major adds also Eb, Eb major adds also Ab etc.
Key signatures put the sharps or flats in this order.
The two meet in the middle:
F# major: F#, G#, A#, B, C#, D#, E# (key with 6 sharps)
Gb major: Gb, Ab, Bb, Cb, Db, Eb, F (key with 6 flats)
I.e. on the piano F# and Gb are the same key, G# and Ab are the same key etc.
I think it makes sense for the other keys, too. Let’s say we had twelve distinct letters, A–L. Here’s a key:
H J K A C E F
First of all, which key is it? We know it starts with H. But is it major or minor? We have to mentally calculate all the intervals, rather than just looking for the sharps and flats.
And calculating the intervals is annoying! You could go 2122212, which doesn’t correspond to major or minor — it’s dorian.
Meanwhile, in current notation, you’d just look at the sharps (C# and F# = D major) and the starting note (E).
Your argument feels quite subjective - the exact opposite argument seems like it could just as valid to somebody else: "looking at sharps is annoying, with 12 distinct notes you can just count the intervals".
I think in an alternate universe where a 12 (or 6+6) notation had historically been used instead of 7+5, the 7+5 option would seem quite unnatural to most people. In any case, starting from a blank slate without historic context, the 7+5 naming scheme just isn't intuitive.
6+6 doesn’t make sense, though — there are 7 notes in any given diatonic scale. You could change the scales around so that they only have 6 notes, but then we’re discussing an entirely different system rather than just a different notation.
Anyway, you can count the intervals now. For example, I know that there are five half steps between F and A#. It is slightly annoying to keep the B and E exceptions in mind, but it’s definitely possible — most people just find it easier to use the sharps and flats as a shortcut.
A 6+6 notation can be used for any kind of scale, including diatonic ones. Just like a 7+5 notation can be used for non-diatonic scales.
A 6+6 system makes more sense in terms of fundamentals - there's the exact same distance between each "full" letter. That doesn't limit what kind of scales you can annotate with the letters at all.
Sure, but the flip side is that every scale now has one repeated letter and at least one sharp or flat. Take C major; in hypothetical 6+6 notation, it becomes
C D E E# F# A# B#
In fact, I think every diatonic scale in 6+6 notation will have three or four sharps/flats.
I just tend to think of scales in terms of numbers and not letters, so the pain of having an "uneven" naming for the notes is worse for me than any pain from no longer having letters line up perfectly with two specific scales. But I can see why the current naming is useful for piano players.
But I've noticed that the people who tend to have the biggest problem with music theory are engineers. I think it comes from an assumption that music should be learned like engineering, where you start with fundamentals, and then build upon them. And then they're horrified to discover that even the fundamentals are arbitrary and approximate.
Most musicians don't start out with fundamentals. We started by choosing an instrument and somehow learning to play it. For the most part, musicians don't care about temperaments. Equal temperament only applies to a handful of instruments that can be kept in tune long enough for it to matter, and that can play the same note, the same way, twice. An orchestra, jazz band, or solo fiddler, have no temperament in any meaningful sense.
Yes, musical notation is a cluster. But it serves its purpose well enough to be useful to support a symbiosis between the composer and the performer. If any composer chooses to try a bespoke notation, they'll discover that nobody will ever perform their composition. It's like a programming language that nobody likes, but that is supported by a job market and labor pool.
My advice for any engineer wanting to get into music is, get an instrument and learn to play it. Preferably a non-electronic one, so you won't be tempted to turn music into a numbers game.